Thermodynamic Forces and Fluxes

Classical thermodynamics is a science of endpoints: it compares equilibrium states but says nothing about the currents that flow between them or the rate at which entropy is generated along the way. Linear irreversible thermodynamics — built by Onsager, Prigogine, de Groot and Mazur — fills that gap by promoting entropy to a local field and writing a balance equation for it. The central object is the entropy production rate $\sigma(\mathbf r,t)\ge0$ , the source term in that balance, which vanishes only in genuine local equilibrium. The decisive structural fact is that $\sigma$ is always a sum of products of fluxes and their conjugate forces: $\sigma=\sum_i J_iX_i$ . This bilinear form is not a modeling choice but a consequence of the local Gibbs relation plus the conservation laws, and it is what dictates the pairing. A flux is a current — of energy, particles, charge, momentum; the conjugate force is the gradient of an intensive field ( $\nabla(1/T)$ for heat, $-\nabla(\mu/T)$ for matter, $E/T$ for charge). At equilibrium every force and every flux vanishes together, and $\sigma=0$ . Close to equilibrium the fluxes are linear in the forces, giving the phenomenological laws $J_i=\sum_j L_{ij}X_j$ . The diagonal coefficients are Fourier's, Fick's and Ohm's laws; the off-diagonal ones are the cross-effects — thermoelectricity, thermodiffusion, electrokinetics. The Onsager reciprocal relations force the matrix to be symmetric, and positivity of $\sigma$ forces it to be positive semi-definite. This article establishes the bilinear structure from first principles, shows how the second law caps the cross-coefficients through $L_{ii}L_{jj}\ge L_{ij}^2$ , and connects the coefficients to equilibrium fluctuations via the Green–Kubo formulas — placing the force/flux framework as the macroscopic face of the fluctuation–dissipation theorem.

Key takeaways

Irreversible processes are described by conjugate pairs of thermodynamic forces $X_i$ (gradients of intensive fields, affinities) and fluxes $J_i$ (currents of heat, matter, charge, momentum).
The local entropy-production density is the bilinear form $\sigma=\sum_i J_iX_i\ge0$ — a statement of the second law for irreversible processes, derived from the local Gibbs relation plus conservation laws.
The bilinear structure uniquely fixes the conjugate force for each flux: heat $\leftrightarrow\nabla(1/T)$ , particles $\leftrightarrow-\nabla(\mu/T)$ , charge $\leftrightarrow E/T$ .
Close to equilibrium, linear laws $J_i=\sum_j L_{ij}X_j$ define the Onsager coefficients; the diagonal entries are Fourier/Fick/Ohm, the off-diagonal ones are cross-effects.
Reciprocity gives $L_{ij}=L_{ji}$ ; positivity of $\sigma$ forces $L_{ii}\ge0$ and $L_{ii}L_{jj}\ge L_{ij}^2$ — the second law caps the cross-coupling.
The coefficients are fixed by equilibrium fluctuations through the Green–Kubo formulas, connecting transport directly to the fluctuation–dissipation theorem.
Key idea: decompose the entropy-production density into a sum of flux times conjugate force, $\sigma=\sum_i J_iX_i\ge0$ ; expand fluxes linearly in forces with Onsager-symmetric, positive-semidefinite coefficients $L_{ij}$ fixed by equilibrium current correlations.

The core insight

Decompose the entropy-production density into a sum of flux times conjugate force, $\sigma=\sum_i J_iX_i\ge0$ ; expand fluxes linearly in forces with Onsager-symmetric, positive-semidefinite coefficients $L_{ij}=L_{ji}$ fixed by equilibrium current correlations.

Where this lives today

The force/flux framework is the language of thermoelectric materials science. Coupling heat and charge gives a $2\times2$ Onsager matrix whose off-diagonal coefficient sets the Seebeck and Peltier effects; the dimensionless figure of merit $ZT=S^2\sigma_e T/\kappa$ that governs device efficiency is built entirely from these coefficients, and the positivity bound $|L_{eq}|\le\sqrt{L_{ee}L_{qq}}$ is exactly why high $ZT$ requires decoupling electronic and thermal transport. The same matrix structure governs coupled transport in membranes and electrolytes — electro-osmosis and streaming potential are reciprocal (Saxén's relations), and cross-diffusion in multicomponent mixtures can drive a species up its own concentration gradient. In computational materials science, the Green–Kubo route is the standard way to obtain transport coefficients: an equilibrium molecular-dynamics simulation records the appropriate current autocorrelation and integrates it, delivering $\kappa$ , $D$ , $\sigma_e$ , and viscosity from the same run. The framework also seeds modern stochastic thermodynamics: the bilinear $\sigma=\sum J_iX_i$ extends to mesoscopic systems where the fluxes are probability currents and the forces are gradients of stochastic entropy, connecting directly to the fluctuation theorems and the entropy production of small driven machines like molecular motors and ion pumps.

The physical picture

Picture two reservoirs joined by a bar: a temperature difference is the "force" that pushes a heat "flux." The entropy production is the rate at which that heat, dumped into the cold side, creates more entropy than it removed from the hot side — proportional to the product of how hard you push (the force) and how much flows (the flux). Just as in mechanics power equals force times velocity, here dissipation equals thermodynamic force times conjugate flux. Equilibrium is the unique state where all forces vanish, all fluxes vanish, and $\sigma=0$ ; departures cost entropy at a rate quadratic in the departure, so the system resists being pushed. The bilinear form $\sum_i J_iX_i$ has a clean geometric reading. The fluxes live in a vector space and the forces in its dual; the coefficient matrix $L_{ij}$ is the metric that maps one to the other. Positive entropy production is the statement that this metric is positive semi-definite. Reciprocity, $L_{ij}=L_{ji}$ , makes it symmetric — the whole framework is the geometry of a symmetric positive-definite bilinear form, with the second law encoded as positivity and microscopic reversibility as symmetry. That reciprocity is best understood through fluctuations. Even in equilibrium, spontaneous microscopic currents fluctuate. The claim that a thermal gradient drives a charge current with the same coefficient that a voltage drives a heat current holds because the underlying fluctuations decay by the same time-reversible microscopic dynamics — Onsager's regression hypothesis, applied to cross-correlations. Linear response, dissipation, and equilibrium noise are three faces of one object: $L_{ij}$ governs the average response (transport), the dissipation (entropy production), and the fluctuations (Green–Kubo). A crucial discipline: the split into "force" and "flux" is fixed only up to the requirement that their product reproduce $\sigma$ . A linear repackaging of the fluxes must be accompanied by the contragredient transformation of the forces, or both the bilinear form and the Onsager symmetry are spoiled. Choosing $\nabla T$ instead of $\nabla(1/T)$ as the thermal force is the classic mistake — it rescales the coefficient by $T^2$ and destroys $L_{ij}=L_{ji}$ .

Figure

The bilinear engine: each flux paired with its conjugate force builds entropy production

Figure

Punchline: the second law is a positivity cone — cross-coupling is capped at the boundary

The core equations

The starting point is the second law in the form Prigogine gave it: the entropy change of an open system splits into a reversible exchange with the surroundings and an irreversible internal production, and only the latter is sign-constrained.

\frac{\mathrm dS}{\mathrm dt}=\frac{\mathrm d_eS}{\mathrm dt}+\frac{\mathrm d_iS}{\mathrm dt},\qquad \frac{\mathrm d_iS}{\mathrm dt}\ge0 \qquad (1)

where: $\mathrm d_eS$ — entropy exchanged with the surroundings (any sign); $\mathrm d_iS$ — entropy produced internally by irreversibility. The second law is $\mathrm d_iS\ge0$ , not $\mathrm dS\ge0$ — an open system can lower its own entropy by exporting more than it makes. Localizing this statement makes entropy a field obeying a continuity equation with a non-negative source.

\frac{\partial s}{\partial t}+\nabla\!\cdot\!\mathbf J_s=\sigma,\qquad \sigma\ge0 \qquad (2)

where: $s$ — entropy density; $\mathbf J_s$ — entropy flux (reversible transport of entropy); $\sigma$ — entropy production density, the source term. This is the local form of the second law. The structural heart of the subject is that the source is bilinear in fluxes and conjugate forces.

\sigma=\sum_i J_i\,X_i\ \ge\ 0 \qquad (3)

where: $J_i$ — thermodynamic flux (energy, matter, charge, momentum current); $X_i$ — conjugate thermodynamic force. The pairing is dictated by the requirement that the product reproduce $\sigma$ ; it is not free. For coupled heat and matter flow, the local Gibbs relation plus conservation laws fixes exactly which gradients are the forces.

\mathbf X_u=\nabla\!\left(\frac1T\right),\qquad \mathbf X_n=-\nabla\!\left(\frac{\mu}{T}\right) \qquad (4)

where: $\mathbf X_u$ — thermal force conjugate to the energy flux $\mathbf J_u$ ; $\mathbf X_n$ — diffusive force conjugate to the particle flux $\mathbf J_n$ . For a charged species one adds $-\nabla(z e\phi/T)$ to recover Ohm's law. Note it is $\nabla(1/T)$ , not $\nabla T$ — the entropy representation demands it. Near equilibrium the fluxes are linear in the forces, and reciprocity makes the coefficient matrix symmetric.

J_i=\sum_j L_{ij}\,X_j,\qquad L_{ij}=L_{ji} \qquad (5)

where: $L_{ij}$ — Onsager kinetic (phenomenological) coefficients. In a magnetic field or rotating frame the symmetry generalizes to $L_{ij}(\mathbf B,\boldsymbol\omega)=L_{ji}(-\mathbf B,-\boldsymbol\omega)$ (Casimir). Positivity of the entropy production, now a quadratic form in the forces, constrains the whole matrix.

\sigma=\sum_{ij}L_{ij}X_iX_j\ge0\ \Rightarrow\ L_{ii}\ge0,\quad L_{ii}L_{jj}\ge\tfrac14(L_{ij}+L_{ji})^2 \qquad (6)

where: $L_{ii}$ — diagonal coefficient (conductivity, diffusivity). Positive semi-definiteness forces non-negative diagonal coefficients and caps the cross-coupling; with reciprocity this reads $L_{ij}^2\le L_{ii}L_{jj}$ . Finally, the coefficients are not merely phenomenological — the Green–Kubo formulas express each as a time integral of an equilibrium current autocorrelation, the precise statement that dissipation is determined by fluctuations.

L_{ij}=\frac{V}{k_B}\int_0^\infty\mathrm dt\,\langle\hat J_i(0)\,\hat J_j(t)\rangle_{\rm eq} \qquad (7)

where: $\hat J_i(t)$ — microscopic flux in equilibrium; $V$ — volume; $\langle\cdot\rangle_{\rm eq}$ — equilibrium average. The symmetry of this integrand under time reversal is a second route to reciprocity (developed in the Green–Kubo article).

Derivation AThe bilinear entropy production from the local Gibbs relation

The goal is to derive the bilinear form $\sigma=\sum_i J_iX_i$ from first principles for coupled heat and matter flow, thereby proving that the identification of forces and fluxes is forced, not chosen. The strategy is to apply the Gibbs relation locally (assuming local equilibrium, so $T$ and $\mu$ are well-defined fields), differentiate the entropy density in time, substitute the conservation laws for energy and particle number, and then separate the result into a divergence (the entropy flux) plus a remainder (the production). The remainder emerges automatically as a sum of current-times-gradient terms, which reads off the conjugate forces.

Step 1. Write the local Gibbs relation per unit volume, $\mathrm ds=\tfrac1T\mathrm du-\tfrac{\mu}{T}\mathrm dn$ , so that $\partial_t s=\tfrac1T\partial_t u-\tfrac{\mu}{T}\partial_t n$ . Why: local equilibrium is the key assumption — it lets us use the equilibrium thermodynamic relation pointwise, with $T(\mathbf r,t)$ and $\mu(\mathbf r,t)$ as smooth fields, even though the system as a whole is out of equilibrium.

Step 2. Insert the conservation laws $\partial_t u=-\nabla\cdot\mathbf J_u$ and $\partial_t n=-\nabla\cdot\mathbf J_n$ (energy and particles are locally conserved, with no sources). Why: the currents enter here; the time derivatives of the densities become divergences of fluxes, which is what lets the entropy balance take continuity form.

Step 3. Use the product rule to split each term into a divergence plus a gradient-times-current piece, collecting the divergences into $\mathbf J_s=(\mathbf J_u-\mu\mathbf J_n)/T$ and leaving the remainder $\sigma=\mathbf J_u\cdot\nabla(1/T)-\mathbf J_n\cdot\nabla(\mu/T)$ . Why: the divergence part is the reversible transport of entropy (the entropy flux), while the remainder is the genuinely irreversible production — the mathematical separation mirrors the physical one.

Step 4. Read off the conjugate forces from the remainder: $\mathbf X_u=\nabla(1/T)$ for the energy flux and $\mathbf X_n=-\nabla(\mu/T)$ for the particle flux, so $\sigma=\mathbf J_u\cdot\mathbf X_u+\mathbf J_n\cdot\mathbf X_n=\sum_i J_iX_i$ . Why: the bilinear form is not imposed — it falls out of the algebra, and with it the unique conjugate force for each flux. This is why using $\nabla T$ instead of $\nabla(1/T)$ is simply wrong: the entropy production dictates the pairing. *Verified: verify_thermodynamic-forces-fluxes.py confirms symbolically that $\tfrac1T\partial_t u-\tfrac{\mu}{T}\partial_t n$ with the conservation laws reduces to $\mathbf J_u\cdot\nabla(1/T)-\mathbf J_n\cdot\nabla(\mu/T)$ , identifying the forces $\nabla(1/T)$ and $-\nabla(\mu/T)$ .*

Go deeper · Derivation B · 2 minPositivity of entropy production caps the cross-coefficients

Derivation BPositivity of entropy production caps the cross-coefficients

The goal is to show that the second law, $\sigma\ge0$ , does far more than fix the sign of the dissipation — it constrains the entire coefficient matrix to be positive semi-definite, forcing $L_{ii}\ge0$ and bounding the cross-couplings by $L_{ii}L_{jj}\ge\tfrac14(L_{ij}+L_{ji})^2$ . The strategy is to write the entropy production as a quadratic form in the forces using the linear laws, observe that non-negativity of a quadratic form for all forces is exactly positive semi-definiteness of the symmetric part of the matrix, and then read off the standard $2\times2$ semi-definiteness conditions. The physical payoff is a hard ceiling on cross-effects like the thermoelectric figure of merit.

Step 1. Substitute the linear laws $J_i=\sum_j L_{ij}X_j$ into $\sigma=\sum_i J_iX_i$ to get $\sigma=\sum_{ij}L_{ij}X_iX_j$ , a quadratic form in the forces. Why: this converts the second-law statement $\sigma\ge0$ from a condition on currents into a condition on the coefficient matrix, which is the object we want to constrain.

Step 2. Require $\sigma\ge0$ for all force vectors $\mathbf X$ . Only the symmetric part $L^{\rm sym}=\tfrac12(L+L^\top)$ contributes to the quadratic form, and non-negativity for all $\mathbf X$ is by definition the positive semi-definiteness of $L^{\rm sym}$ . Why: a quadratic form's sign depends only on the symmetric part of its matrix; demanding non-negativity for every possible driving is what turns the second law into a matrix inequality.

Step 3. Read off the semi-definiteness conditions: every diagonal entry $L_{ii}\ge0$ (so conductivity, diffusivity, thermal conductivity are all non-negative), and every $2\times2$ principal minor is non-negative, $L_{ii}L_{jj}\ge\tfrac14(L_{ij}+L_{ji})^2$ . With Onsager symmetry $L_{ij}=L_{ji}$ this becomes $L_{ij}^2\le L_{ii}L_{jj}$ . Why: positive semi-definiteness of a symmetric matrix is exactly the non-negativity of its leading principal minors — the diagonal conditions give the sign of transport coefficients, the minor conditions cap the cross-coupling.

Step 4. Interpret the bound. A material cannot have an arbitrarily strong cross-effect: $|L_{ij}|\le\sqrt{L_{ii}L_{jj}}$ . For thermoelectrics this is why the figure of merit $ZT$ has a ceiling set by the diagonal transport, and why raising it requires decoupling electronic and thermal conductivity. Why: the second law is not just bookkeeping — it is a design constraint, limiting how efficiently any coupled transport device can convert one current into another. *Verified: verify_thermodynamic-forces-fluxes.py draws 200 random $2\times2$ matrices, confirms that $\sigma\ge0$ for all forces is equivalent to positive semi-definiteness of the symmetric part, and checks $L_{ii}\ge0$ and $L_{ii}L_{jj}\ge\tfrac14(L_{ij}+L_{ji})^2$ in every semi-definite case.*

Limiting cases

The force/flux framework specializes to the familiar transport laws and breaks outside its regime:

Case	Result	Meaning
Heat conduction	$\mathbf J_q=-\kappa\nabla T$ , $X_q=\nabla(1/T)$	Fourier's law; $L_{qq}=\kappa T^2$ relates the coefficient to the conductivity $\kappa$ .
Particle diffusion	$\mathbf J_n=-D\nabla n$ , $X_n=-\tfrac1T\nabla\mu$	Fick's law; the Einstein relation $D=\mu_{\rm mob}k_BT$ links it to mobility.
Charge transport	$\mathbf J_e=\sigma_e\mathbf E$ , $X_e=\mathbf E/T$	Ohm's law; $L_{ee}=\sigma_e T$ ties the coefficient to conductivity.
Far from equilibrium	$J_i=\sum_j L_{ij}X_j+\tfrac12\sum_{jk}L_{ijk}X_jX_k+\cdots$	Linear LIT fails; nonlinear terms appear, Onsager symmetry no longer guaranteed, and steady states need extended thermodynamics.

Common trapThe force/flux split is fixed only up to a contragredient transformation

The decomposition into "force" and "flux" is not unique — it is fixed only by the requirement that their product reproduce $\sigma=\sum J_iX_i$ . If you redefine the fluxes by a linear map $J'=AJ$ , you must transform the forces by the contragredient map $X'=(A^{-1})^\top X$ to preserve both the entropy production and the Onsager symmetry $L\to ALA^\top$ . Mixing fluxes without the matching inverse-transpose on forces destroys reciprocity. The most common concrete error is using $\nabla T$ instead of $\nabla(1/T)$ as the thermal force: this rescales the coefficient by $T^2$ and misaligns the cross-coefficients so that $L_{eq}\ne L_{qe}$ . Reciprocity is a statement about the entropy-conjugate variables, and only about them.

Worked examples

Example 1 — Local entropy production in a heated copper bar. This computes the entropy generated per unit volume by steady heat conduction, using real numbers.

Step 1. Take copper with thermal conductivity $\kappa=400\ \mathrm{W/m/K}$ , at $T=300\ \mathrm K$ , with a temperature gradient $\nabla T=50\ \mathrm{K/m}$ . Fourier's law gives the heat flux $\mathbf J_q=-\kappa\nabla T=-400\times50=\mathbf{-2.0\times10^{4}\ W/m^2}$ (heat flows down the gradient).

Step 2. The conjugate force is $X_q=\nabla(1/T)=-(1/T^2)\nabla T=-(50/300^2)=-5.56\times10^{-4}\ \mathrm{K^{-1}\,m^{-1}}$ .

Step 3. The local entropy production is $\sigma=\mathbf J_q\cdot X_q=(-2.0\times10^4)\times(-5.56\times10^{-4})=\kappa(\nabla T)^2/T^2=400\times2500/90000=\mathbf{11.1\ W/(K\,m^3)}$ .

Step 4. This is manifestly positive because it is $\kappa(\nabla T)^2/T^2$ — a positive coefficient times a squared gradient. Every cubic metre of the bar generates $11.1$ watts-per-kelvin of entropy as heat flows through it, and the entropy production vanishes only when the gradient does. The mapping to the Onsager coefficient is $L_{qq}=\kappa T^2=3.6\times10^7$ , so that $J_q=L_{qq}X_q$ reproduces Fourier's law.

Example 2 — The Einstein relation as a one-flux special case. This shows how demanding zero net current in equilibrium fixes the diffusion–mobility ratio.

Step 1. For a single diffusing species in an external potential, the flux is $J=\mu_{\rm mob}Fn-D\,\partial_x n$ — a drift term (mobility times force times density) plus a diffusive term (Fick's law). This is the linear law $J=\sum L_{ij}X_j$ written out.

Step 2. In sedimentation equilibrium the net flux vanishes, $J=0$ , even though both drift and diffusion are present. The equilibrium density is the barometric (Boltzmann) law $n\propto e^{-U/k_BT}$ , so $\partial_x n=-(F/k_BT)n$ with $F=-\partial_x U$ .

Step 3. Substituting into $J=0$ : $\mu_{\rm mob}Fn-D(-(F/k_BT)n)\cdot(-1)=0$ collapses (the force $F$ cancels) to $D=\mu_{\rm mob}k_BT$ . For a mobility $\mu_{\rm mob}=2.5\times10^{11}\ \mathrm{s/kg}$ at $T=300\ \mathrm K$ , this gives $D=\mu_{\rm mob}k_BT=\mathbf{1.04\times10^{-9}\ m^2/s}$ .

Step 4. The single coefficient cancels, leaving the fluctuation–dissipation identity $D=\mu_{\rm mob}k_BT$ — diffusion (a fluctuation-driven current) and mobility (a dissipative response) are locked together by $k_BT$ . This is the simplest instance of the deep statement that the force/flux coefficients are fixed by equilibrium fluctuations.

Origins · 1 mintransport coefficient and how this was discovered

transport coefficient

kinetic coefficients organized by Onsager (1931), Prigogine, de Groot & Mazur (1962) · A transport coefficient is the proportionality constant relating a flux to its driving force — the thermal conductivity

\kappa

in Fourier's law, the diffusion constant

D

in Fick's law, the electrical conductivity

\sigma_e

in Ohm's law, and the off-diagonal coefficients coupling heat to charge or matter. Lars Onsager's 1931 reciprocal relations first organized these coefficients into a symmetric matrix, and Ilya Prigogine together with Sybren de Groot and Peter Mazur built the systematic theory of linear irreversible thermodynamics around them in the mid-twentieth century, culminating in de Groot and Mazur's 1962 monograph. The transport coefficients are simultaneously the average response (how much current a force drives), the dissipation (how much entropy is produced), and the equilibrium fluctuations (through the Green–Kubo formulas) — three faces of one object, which is why measuring one property of a material constrains the others.

History

1822

— Fourier publishes the law of heat conduction, the prototype flux–force relation

1855

— Fick formulates the diffusion law by analogy with Fourier

1905

— Einstein derives the diffusion–mobility relation, tying transport to fluctuations

1931

— Onsager proves the reciprocal relations, organizing the transport coefficients into a symmetric matrix

1962

— de Groot and Mazur publish Non-Equilibrium Thermodynamics, the systematic theory of linear irreversible thermodynamics

Going further

References. R. K. Pathria & P. D. Beale, Statistical Mechanics, 3rd ed. (Elsevier, 2011), Ch. 15 · M. Kardar, Statistical Physics of Particles (Cambridge, 2007), Ch. 3 · F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965), Ch. 15 · S. R. de Groot & P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, 1962) · L. Onsager, Phys. Rev. 37, 405 (1931); 38, 2265 (1931) · L. D. Landau & E. M. Lifshitz, Statistical Physics Part 1, §120 · J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity (Oxford, 2006).

Provenance. Structure and equations follow de Groot & Mazur, Onsager (1931), and standard texts (Pathria & Beale Ch. 15, Reif Ch. 15, Landau–Lifshitz §120). A verifier recomputes: (1) the bilinear decomposition of $\sigma$ from the local Gibbs relation and conservation laws, identifying the forces $\nabla(1/T)$ and $-\nabla(\mu/T)$ ; (2) the positivity conditions $L_{ii}\ge0$ and $L_{ii}L_{jj}\ge\tfrac14(L_{ij}+L_{ji})^2$ over 200 random matrices; (3) the heat-conduction entropy production $\sigma=\kappa(\nabla T)^2/T^2=11.1\ \mathrm{W/(K\,m^3)}$ with $L_{qq}=\kappa T^2$ ; (4) the Einstein relation $D=\mu_{\rm mob}k_BT=1.04\times10^{-9}\ \mathrm{m^2/s}$ from $J=0$ ; (5) the contragredient invariance of $\sigma$ and the Onsager symmetry. The figure of merit $ZT=S^2\sigma_e T/\kappa$ is the standard thermoelectric definition.

Builds on: The Einstein Relation · The Fluctuation-Dissipation Theorem · The Langevin Equation Leads to: Green-Kubo relations · The Fokker-Planck Equation · The Diffusion Equation

How do I know which quantity is the "force" and which is the "flux"?

The bilinear entropy production decides it. Derive $\sigma$ from the local Gibbs relation plus the conservation laws, and it comes out as a sum of current-times-gradient terms. Whatever multiplies a current in that expression is the conjugate force. For heat this gives $\mathbf J_q\cdot\nabla(1/T)$ , so the force is $\nabla(1/T)$ ; for particles it is $-\nabla(\mu/T)$ ; for charge it is $\mathbf E/T$ . The flux is the physical current (of energy, matter, charge). You are not free to choose the pairing — the requirement that the product equal $\sigma$ fixes it, and only this pairing has the Onsager symmetry.

Why does the second law appear as positivity of a quadratic form?

Because near equilibrium the fluxes are linear in the forces, $J_i=\sum_j L_{ij}X_j$ , so the entropy production $\sigma=\sum_i J_iX_i$ becomes the quadratic form $\sigma=\sum_{ij}L_{ij}X_iX_j$ . The second law demands $\sigma\ge0$ for every possible set of driving forces, and a quadratic form that is non-negative for all arguments is by definition positive semi-definite. So the second law is exactly the statement that the (symmetric part of the) transport matrix is positive semi-definite — which forces $L_{ii}\ge0$ and caps the cross-coefficients. Dissipation is a non-negative quadratic form in the forces, vanishing only at equilibrium.

Can a system really lower its own entropy without breaking the second law?

Yes, and the framework makes it precise. The entropy change splits as $\mathrm dS=\mathrm d_eS+\mathrm d_iS$ ; the second law constrains only the internal production $\mathrm d_iS\ge0$ . If the system exports entropy to its surroundings faster than it produces it ( $\mathrm d_eS<0$ and $|\mathrm d_eS|>\mathrm d_iS$ ), its own entropy $\mathrm dS$ falls. This is exactly how crystals grow, how convection cells organize, and how living organisms maintain low-entropy structure — by continuously dumping entropy into the environment. Prigogine's dissipative structures are the systematic study of such entropy-exporting steady states.

What is the difference between the entropy production and the entropy flux?

The entropy production $\sigma\ge0$ is the irreversible source term — entropy created inside a volume element by dissipative processes, always non-negative. The entropy flux $\mathbf J_s=(\mathbf J_u-\mu\mathbf J_n)/T$ is the reversible transport of entropy across boundaries, and it can have either sign. In the balance $\partial_t s+\nabla\cdot\mathbf J_s=\sigma$ , the flux moves entropy around while the production makes it. Confusing the two is a common trap: a steady state has $\partial_t s=0$ , so all the entropy produced internally is exactly exported by the flux, $\int\sigma\,\mathrm dV=\oint\mathbf J_s\cdot\mathrm d\mathbf A$ .

How are the Onsager coefficients connected to equilibrium fluctuations?

Through the Green–Kubo formulas: each coefficient is a time integral of an equilibrium current autocorrelation, $L_{ij}\propto\int_0^\infty\langle\hat J_i(0)\hat J_j(t)\rangle_{\rm eq}\,\mathrm dt$ . This is the quantitative content of Onsager's regression hypothesis — spontaneous equilibrium currents decay by the same macroscopic laws that describe driven transport, so the integrated autocorrelation of the fluctuations equals the transport coefficient. It means the dissipative response of a material is completely determined by its equilibrium noise, the defining statement of the fluctuation–dissipation theorem, and it is why an equilibrium simulation (which never applies a field) can predict conductivity, diffusion, and viscosity.

Why is the thermoelectric figure of merit ZT bounded?

Because the second law caps the cross-coefficient. Positivity of entropy production requires $L_{eq}^2\le L_{ee}L_{qq}$ , i.e. the thermoelectric coupling cannot exceed the geometric mean of the electrical and thermal transport. The figure of merit $ZT=S^2\sigma_e T/\kappa$ is built from these coefficients, and this bound limits how large it can be for given diagonal transport. That is precisely why the search for better thermoelectrics is a search for materials that decouple electronic conductivity from thermal conductivity — phonon-glass electron-crystal materials — rather than simply cranking up the cross-effect, which the second law forbids.